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laser-induced electron-phonon nonequilibrium: A
first-principles study
Emile Bévillon, Jean-Philippe Colombier, Vanina Recoules, Razvan Stoian
To cite this version:
nonequilibrium: a first-principles study
E. B´evillon,1 J.P. Colombier,1, ∗ V. Recoules,2 and R. Stoian11
Laboratoire Hubert Curien, UMR CNRS 5516, Universit´e de Lyon, Universit´e Jean-Monnet 42000, Saint-Etienne, France
2
CEA-DIF, 91297 Arpajon, France (Dated: April 3, 2014)
The electronic behavior of various solid metals (Al, Ni, Cu, Au, Ti and W) under ultrashort laser irradiation is investigated by means of density functional theory. Successive stages of extreme nonequilibrium on picosecond timescale impact the excited material properties in terms of optical coupling and transport characteristics. As these are generally modelled based on the free electron classical theory, the free electron number is a key parameter. However, this parameter remains unclearly defined and dependencies on the electronic temperature are not considered. Here, from first-principles calculations, density of states are obtained with respect to electronic temperatures varying from 10−2K to 105
K within a cold lattice. Based on the concept of localized or delocalized electronic states, temperature dependent free electron numbers are evaluated for a series of metals covering a large range of electronic configurations. With the increase of the electronic temperature we observe strong adjustments of the electronic structures of transition metals. These are related to variations of electronic occupation in localized d-bands, via change in electronic screening and electron-ion effective potential. The electronic temperature dependence of nonequilibrium density of states has consequences on electronic chemical potentials, free electron numbers, electronic heat capacities and electronic pressures. Thus electronic thermodynamic properties are computed and discussed, serving as a base to derive energetic and transport properties allowing the description of excitation and relaxation phenomena caused by rapid laser action.
PACS numbers: 81.40.-z,79.20.Ds,72.15.Lh
I. INTRODUCTION
The dynamics of laser-excited materials is an area of in-tense research as diagnostics of laser-matter experiments can be discussed by back-tracking the transient proper-ties of the irradiated material. Particularly, the primary phenomena of transient electronic excitation and energy transport are of utmost importance. Irradiating a metal by a short laser pulse (∼ 100 fs) can lead to a signifi-cant rise of the electronic temperature with respect to the ionic lattice as the energy of the laser pulse can be deposited before the material system starts dissipating energy by thermal or mechanical ways. The electronic excitation can affect both electronic and structural prop-erties of the solid, impacting optical coupling, transport and phase transitions. The confinement of the absorbed energy at solid density pushes the matter into an extreme nonequilibrium state and new thermodynamic regimes are triggered. The interplay between the ultrafast excita-tion and the material response still requires a comprehen-sive theoretical description for highly excited solid ma-terials including in particular the excitation-dependent band structure evolution as this influence the response to laser action.1 Recent advances in studying processes
induced by short laser pulses have revealed the deter-minant role of primary excitation events. Their accurate comprehension is necessary to correctly describe ultrafast structural dynamics,2,3 phase transitions,4,5
nanostruc-ture formation,6 ablation dynamics,7,8 or strong shock
propagation.9 In such nonequilibrium conditions,
con-duction electrons participating to energy exchange are expected to evolve in time, depending on the excitation degree.10 They largely determine the material transient
properties and transformation paths. In this context, they are a crucial parameter required to describe com-plex ultrafast phenomena involving relaxation of excited states. Particularly, pure electronic effects (population and band distribution) determining transient coefficients before structural transitions set in are of interest and we will follow excitation influence in the form of nonequilib-rium electronic temperature.
At the very beginning of the irradiation process, ex-cited electrons are unhomogeneously distributed within the electronic band structure of the materials. By colli-sions and energy transfer, they fastly reach a Fermi-Dirac distribution. The electronic subsystem is thermalized and the concept of electronic temperature (Te) can be
applied. Mueller et al. have recently shown that the elec-tron subsystem thermalizes within a characteristic time τ in the range of tens of femtoseconds for Telarger than
104K.11 Then, a first relaxation channel is the energy
transfer between electronic and vibrational excitations. This is commonly described by a two temperature model (TTM)12,13 based on the assumption that the
occupa-tion of electronic and phonons states can be separately described by two effective temperatures, the electronic Te
and the ionic temperature Ti. In standard approach,
occurs in the picosecond timescale.14Thus, there exists a
period of time where the ionic temperature remains low while electrons fastly reach a thermalized state of high electronic temperature. In this case, strong alteration of electronic properties preceeds structural transformation, with consequences on the efficiency of energy deposition. Such nonequilibrium states can be modelled in the frame-work of first-principles calculations, by interrogating the electronic influence at various degrees of electronic heat-ing, while disregarding in a first approximation, the ionic temperature effects.
Experimentally, excited solids in steady-state cannot be created and thermal nonequilibrium data in these par-ticular conditions are difficult to be determined from inte-grated or time-resolved measurements.15–17In such com-plex conditions, simple estimations and models are used to access the behavior of intrinsic material properties. A strong need then exists to perform multiscale calcula-tions both in space and in time, capable of replicating the observed behaviors and to predict material response under excitation. Most of the macroscopic models and behavior laws are based on the picture of free electrons commonly used to describe metals or even dielectric dy-namics under laser irradiation.1,3,13,18 Ab initio
micro-scopic calculations can supply macromicro-scopic approaches (optical, thermal, hydrodynamical or mechanical) with implicit dependencies on material properties and elec-tronic band structures. Since the density of free electrons ne is not an observable variable in a quantum
mechani-cal perspective, its dynamics in thermally excited solids remains poorly explored whereas the transport param-eters and measurable dynamics are commonly depicted and fitted by laws depending on ne.
It has already been shown in previous works that elec-tronic structures determine the thermodynamic functions and scattering rates of the heated electron subsystem.19
This work analyses and extends the largely used elec-tronic thermodynamic properties derived from the free electron gas model by interrogating the evolution of the free carriers. This model, based on the assumption of free and non-interacting electrons, works satisfactory in case of simple metals (Na, Mg, Al...). However it can-not encompass the complexity observed for transition metals, where d-electrons with a higher degree of local-ization than sp-electrons can still participate to optical processes. This indicates a potentially important role of electron confinement within more or less diffuse orbitals. Moreover, it has already been shown that the increase of the electronic temperature strongly affects the shape of the d-band.5 Under such strong modifications of the
electronic systems, it is important to extract from cal-culations an effective free electron number per atom Ne
classically defining ne= Neni. This effective parameter
can have importance whenever experimental optical or thermal properties are derived and used to extract other parameters such as temperature, stress or conductivity from a nonequilibrium solid. The objective of the inves-tigations presented here is to quantify the effects of
ther-mal activation energy ∼ kBTearound Fermi energy on Ne
consistently with a rigorously calculated band-structure accounting for Fermi smearing and d-band shifting within the range of 0.01 ≤ Te≤ 105K.
We report results from a systematic study on DOS en-ergy broadening performed on a free electron like metal (Al) and on transition metals (Ni, Cu, Au, W and Ti), some of them with noble character. Section II is de-voted to the calculation and procedure details. In sec-tion III, DOS dependence on the electronic temperature is discussed. We focus on the observable energetic shift and narrowing of the d-band and the implications on the chemical potential. Finally, in order to obtain the elec-tron density relevant to light absorption, heat flux, or mechanical stress induced by electronic heating, an esti-mation of the number of free electron per atom, based on delocalized states considerations, is calculated at all electronic temperatures and discussed for all metals in Section IV. Concluding remarks on the effects of ne
evo-lution on energetic and transport parameters, especially on electronic pressure and electronic heat capacity are made in section V.
II. CALCULATIONS DETAILS
Calculations were done in the framework of the den-sity functional theory (DFT),20,21 by using the Abinit
package22 which is based on a plane-waves description
of the electronic wave functions. Projector augmented-waves atomic data23–25(PAW) are used to model nucleus
and core electrons. The generalized gradient approxi-mation (GGA) in the form parameterized by Perdew, Burke and Ernzerhof26 or the local density
approxima-tion (LDA) funcapproxima-tional developed by Perdew and Wang27
are considered for the exchange and correlation func-tional. Semicore electronic states are included in PAW atomic data of Ti and W as they significantly improve the description of material properties. The Brillouin zone was meshed with Monkhorst-Pack method,28with a 30 ×
30 × 30 k-point grid. From the studied metals, only Ni is expected to have ferromagnetic properties, but our cal-culations showed that magnetic properties vanish above Te = 3 × 103K, thus, all calculations were done without
using spin-polarized methods. To ensure high accuracy of calculations, lattice parameters were relaxed up to the point where stress goes beyond 10−4eV/˚A, with a cutoff
energy of 40 Ha.
Al, Ni, Cu, Au, Ti, W cristallize in different phases de-pending on the environment conditions. Here, we focus on cristal structures adopted by metals at ambiant con-ditions, namely: face-centered cubic (FCC, space group F m¯3m, 225) structure for Al, Ni, Cu and Au; hexagonal close-packed (HCP, space group P 63/mmc, 194)
lat-TABLE I. Electronic structure of atoms and theoretical, experimental and relative error (%) of lattice parameters (˚A) and bulk moduli (GPa) of metal phases at ambiant conditions.
Elt XC Functionals Elec. Struc. Chem. Struc. lth lexp Rel. Err. Bth Bexp Rel. Err.
Al GGA 3s2 3p1 FCC 4.04 4.05 -0.4 79 81 -2 Ni GGA 3d8 4s2 FCC 3.51 3.52 -0.3 192 191 1 Cu GGA 3d10 4s1 FCC 3.64 3.61 0.6 142 133 6 Au LDA 5d10 6s1 FCC 4.05 4.08 -0.7 195 167 14 Ti GGA 3s2 3p6 4s2 3d2 HCP (a,b) 2.93 2.95 -0.6 112 114 2 HCP (c) 4.66 4.69 -0.6 W GGA 5s2 5p6 4f14 5d4 6s2 BCC 3.18 3.17 0.7 295 296 -5
tice parameters and bulk moduli using Birch-Murnaghan equation of states. A good agreement is found between our calculated values and experimental data, that are provided in Table I. Some differences are noticeable be-tween computed and experimental bulk moduli,31–33
es-pecially when zero-point phonon effects are not taken into account.31This confirms the reliability of the used PAW
atomic data. The theoretical lattice parameters com-puted at this step are then used to calculate Tedependent
DOS.
To model laser irradiation effects, we consider timescales where the electrons are thermalized and their distribution can be described by electronic temperatures. Calculations were done with Te ranging from 10−2K to
105K while T
iremains equal to 0K. A number of 40 bands
per atom is used to ensure a maximum occupation be-low 10−4 electrons of the highest energy band at 105K.
Te dependent DFT calculations are performed following
the generalization of the Hohenberg and Kohn theorem on many-body systems to the grand canonical ensemble as proposed by Mermin.29 The finite electronic
temper-ature is taken into account by considering a Fermi-Dirac distribution function applied to the Kohn-Sham eigen-states, ensuring a single thermalized state of electrons during the self-consistent field cycle. This involves a Te
dependent electronic density and an electronic entropy part in the free energy potential with implicit and explicit dependencies.22,30 Equilibrium electronic density at a
fi-nite electronic temperature is determined by minimizing the free energy, the variational functional here, resulting in a Tedependent electronic structure.
III. Te EFFECT ON DENSITY OF STATES
The following discussion is based on the precise de-termination of the electronic density of states of all the considered metals. Calculations are performed at twelve different electronic temperatures, from 10−2 to 105K.34
The DOS and associated Fermi-Dirac electronic distribu-tion funcdistribu-tions of the discussed metals are shown in Fig. 1 at the electronic temperatures of 10−2, 104and 5 × 104
K. Here, for simplicity, the beginning of the valence band of the DOS was set at 0 eV for each Te. According to
this representation, the number of valence electrons Nv e
can be expressed as:
Nv e =
Z ∞ 0
g(ε, Te)f (ε, µ, Te) dε, (1)
where g(ε) is the DOS and f (ε, µ, Te) is
the Fermi-Dirac distribution (f (ε, µ, Te) =
{exp[(ε − µ(Te))/(kbTe)] + 1}−1).
If we first focus the discussion on the DOS obtained at Te=0K, we can notice that they are similar to
pre-vious works.5,19,35 For Al, the DOS adopts the shape of
square root function of the energy, characteristic for a free electron like metal. With transition metals, the d-band appears with a typical d-block having a much higher den-sity. This d-block is filled or almost filled in case of Ni, Cu and Au, while the filling is roughly 1/3 and 1/2 in the case of Ti and W as showed by the location of the Fermi energy. Generally speaking, the d-bands are narrow in the case of Ni, Cu and Au since almost all d-states are filled, which leads to a weak and non-directional charac-ter of the d-bonding.36 At the opposite, they are much
more expanded with the presence of pseudo-gaps in the case of Ti and W, exhibiting a stronger and more direc-tional d-bonding.36
A. Shift and shrinking of thed-block
Density of states exhibit different dynamics when the electronic temperature increases. In the case of Al for example, the DOS is almost insensitive to Te. This
con-stant behavior of the electronic structure was already noticed in Ref. [5]. On an other hand, transition met-als exhibit more complex DOS due to the presence of d-bands. Metals with d-block fully or almost fully occu-pied by electrons (Ni, Cu and Au) exhibit a shrinking and a strong shift of the d-block toward lower energies when Te increases. On the contrary, metals with
par-tially filled d-block (Ti and W) display an expansion and a shift toward higher energies of their d-block when Te
is increased. In order to quantify these phenomena, we show in Fig. 2 the relative change of the d-block center [∆εd(Te) = εd(Te) − εd(0)] as well as the relative change
of the d-block width [∆Wd(Te) = Wd(Te) − Wd(0)] with
0.0 0.2 0.4 0.6 0.8 0.0 1.5 3.0 4.5 6.0 0.0 2.0 4.0 6.0 8.0 0.0 1.2 2.4 3.6 4.8 0.0 1.0 2.0 3.0 4.0 0 4 8 12 16 20 0.0 1.3 2.6 3.9 5.2 0K 10000K 50000K (f) W, BCC (e) Ti, HCP (d) Au, FCC (c) Cu, FCC (b) Ni, FCC (a) Al, FCC D en si ty o f e le ct ro ni c st at es [s ta te s/ eV /a to m ] Energy [eV]
FIG. 1. (color online). Electronic density of states (solid lines), associated Fermi-Dirac distribution functions (dotted lines) and corresponding electronic chemical potential (dashed lines) for all studied metals. Data for the following electronic temperatures are shown: 10−2K (black), 104
K (red) and 5 × 104
K (blue) curves.34
rectangular band model, whose sides are evaluated from the side slope of the electronic density surrounding the d-block. Then, the d-block center and width are estimated as εd(Te) = (εrd+ ε
l
d)/2 and Wd(Te) = εrd− ε l
d, where r
and l superscripts correspond to the right and left sides of the rectangle. 0 2 4 6 8 10 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 (b) (a) W d [e V] Ni Cu Au Ti W d [e V] Electronic temperature [104 K]
FIG. 2. (color online). Relative changes of the d-block center (a) and width (b) with the electronic temperature, for all studied transition metals.
The d-block modifications observed here for Au were already reported in Ref. [5], with an explanation of the phenomenon based on changes of the electronic screening. When an electronic temperature is applied, the depopu-lation of 5d-block leads to a decrease of the electronic screening which makes the effective electron-ion poten-tial more attractive.5 The consequence is a global shift
of the electronic states toward lower energies. A similar effect likely occurs in the case of Ni and Cu even if the depopulation is now concerning 3d-electrons instead of 5d as in Au. Considering the shift toward higher ener-gies and extension of the d-block in case of Ti and W, one should expect an increase of the screening effect with the augmentation of Tefor these two metals. To validate this
assumption, the change of the number of d-electrons ∆Nd
and the changes of Hartree energies ∆EHa as a marker
of the changes of electronic localization have been eval-uated and are presented in Fig.3. We specify that the concept of electronic localization refers to a certain de-gree of spatial concentration of the charge density. The relevance of these observations are discussed below.
Except the case of Al, all considered metals have elec-tronic configurations in the form of (n-1)dxnsy. Since
the main quantum number is one of the dominant pa-rameter characterizing the diffuse nature of an orbital,37
in--1.8 -1.5 -1.2 -0.9 -0.6 -0.3 0.0 0.3 0 2 4 6 8 10 -60 -45 -30 -15 0 15 30 (a) N d (p er a to m ) (b) Al Ni Cu Au Ti W E H a [e V/ at om ] Electronic temperature [104 K]
FIG. 3. (color online). Changes of the number of d-electrons (a) and changes of the Hartree energy (b) with respect to the electronic temperature.
sight, we computed the number of d-electrons from inte-grations of angular-momentum projected DOS, and we plotted ∆Nd as a function of Te in Fig. 3a. As
ex-pected, ∆Nd decreases for Ni, Cu and Au. Te
depopu-lates the d-band, which leads to a decrease of the elec-tronic screening as discussed previously. At the opposite, ∆Nd increases for Ti and W. This is a consequence of
partially filled d-block, since electronic excitation depop-ulates both sp-bands and the bottom part of the d-band and populates the top part of the d-band. Semicore elec-tronic states also undergo an elecelec-tronic depopulation at Teabove 5 × 104K. For example, at 105K, the
depopula-tion reaches 0.3 electrons for 3p-semicore electronic states of Ti and 0.4 electrons for 4f -semicore electronic states of W. Consequently, the total number of d-electrons in-creases in agreement with a strengthening of the elec-tronic screening at least up to 5 × 104K, where the
de-population of semicore states seems to moderate the ef-fect, as we can see on Fig. 2 with a decrease of shifts and width changes. Finally, the behavior of Al is particular with an increase of Nd and no effects on the density of
states. This is related to the occupation of a high en-ergy d-band that does not contribute to the electronic localization.
However, DOS angular-momentum projection meth-ods suffer from some drawbacks. Firstly, they are per-formed on spheres centered on atoms and there is always some intersphere informations lost during the process. Consequently, some electronic states and some electrons are not accounted for. Secondly, projections on angular-momentum do not allow distinctions between 3d, 4d or 5d bands. Then, the number of d-electrons computed
can be the summation of electrons from localized (n-1)d band with nd or even (n+1)d delocalized bands. Thus, the variation of d-electron numbers has some uncertain-ties, but still provides information about trends. In or-der to get a more precise perspective of how electronic screening is affected by ∆Nd, we rely on Hartree energy.
The changes of Hartree energies with respect to Te are
plotted in Fig. 3b. We recall that this energetic term is the Coulomb repulsive self-energy of the electronic den-sity: EHa = 12
RRn(r)n(r′)
|r−r′| drdr′. It corresponds to the
global electron-electron interaction. In a perfectly homo-geneous spatial distribution, this quantity would reach a minimum value, however, constrained by the electronic structure, electrons are not homogeneously distributed. Specifically, semicore states and (n-1)d-bands strongly concentrate the charge density, which generates an tronic screening of the nucleus. Variations of the elec-tronic occupation of these elecelec-tronic states with Te
im-pact Hartree energies, which can be related to a change of electronic screening. Thus, the evolution of ∆EHa with
Te is an indicator for the gain or the loss of electronic
localization. This will be mainly the consequence of Nd
changes due to the spatial confinement of the (n-1)d-band we discussed above. The evolution of ∆EHa also reflects
the change of electronic screening of the ions. On Fig. 3, we can notice a good correlation between ∆Nd and
∆EHa with Te at least up to temperatures of 5 × 104K.
When the number of d-electrons decreases there is a loss of electronic localization. This leads to a decrease of EHa
which signals a decrease of electronic screening. The op-posite phenomenon occurs when the Nd increases, with
a gain of electronic localization leading to an increase of EHacorrelated to an increase of electronic screening.
Fi-nally, ∆EHa of Al remains roughly equal to zero, while
the electronic occupation of high energy bands increases. It confirms that high energy bands, including d-bands, are sufficiently delocalized to be considered as uneffec-tive on electronic localization.
From EHa and Nd changes, some unexpected
behav-iors are also observed. Firstly, ∆Nd is correlated to
∆EHa in the case of Cu and Ni. The stronger is the
decrease of ∆Nd, the stronger is the decrease of ∆EHa.
However, this rule is no longer available when consid-ering Au. This is due to the fact that d-electrons be-long to the 5d-orbitals in the case of Au whereas they belong to 3d-orbitals in the case of Cu and Ni. Associ-ated to more diffuse orbitals, d-electrons of Au are al-ready less localized than those of Cu and Ni, and it leads to a lower loss of localization when d-band is depopu-lated with the increase of Te. In other words, ∆EHa is
lowered in the specific case of Au since its d-electrons are already less localized. Secondly, ∆EHa is not
cor-related to ∆Nd in case of Ti and W at electronic
The strong evolution of Hartree energy for W originates from the significant change of electronic screening gener-ated by the depopulation of highly localized 4f -electrons. We assume here that semicore electrons are fully ther-malized with valence electrons even if semicore thermal-ization timescales are difficult to estimate. Being low in energies, these semicore states are not directly excited by laser irradiation and thermalization time is dependent on electron-electron collision frequency.11,38At 105K, Fisher et al.38 estimate about 1-10 sucessfull impact
probabili-ties during irradiation timescale, for binding energies of semicore electronic states similar to the ones considered here. This suggests a short thermalization time but accu-rate description is beyond the scope of the present work. Generally speaking, we note a relatively good agree-ment between ∆Ndand ∆EHaon one side, and ∆εd and
∆Wd on the other side. For Ni, Cu and Au, the decrease
of ∆Ndleads to a loss of electronic localization that
gen-erates a decrease of ∆EHa. The effect can be related
to the decrease of the electronic screening and thus to the increase of the global electron-ion effective potential that shifts electronics states of these metals toward lower energies. This shift applies non-homogeneously on the d-block since bottom d-block is less affected by depopu-lation and thus by changes of electronic screening than the top part, as can be seen on Fermi-Dirac distributions on Fig. 1(b-d). As a result the d-block is shrinked, and ∆εd and ∆Wd decrease. Shifts also apply to electronic
states of higher energies, as shown for Cu on Fig. 4(a,b). For Ti and W, the increase of ∆Nd leads to an increase
of ∆EHa. The corresponding gain in electronic
localiza-tion produces a stronger electronic screening. As a result, the electron-ion effective potential is less attractive and bands are shifted toward higher energies. It also applies inhomogeneously to the d-block with its bottom states less affected by changes of electronic population than its top states, as can be seen on Fermi-Dirac distributions on Fig. 1(e,f). As a consequence, the d-block extends and is shifted toward higher energies, and ∆εd and ∆Wd
in-crease. Higher energy electronic states are also affected by this increase of electronic screening, as we can note with the shift of other bands toward higher energies for Ti on Fig. 4(c,d). This discussion synthesizes Teeffects
on DOS for a range of representative metals, with various possible impacts on electronic properties.
B. Electronic distributions
As discussed above, the electronic distribution is en-sured by the Fermi-Dirac function characterized by the electronic chemical potential µ(Te). As already showed
by Lin et al.,19the electronic chemical potential exhibits
strong variations, depending on the material studied. More precisely, these variations are related to the assym-metric distribution of the density of electronic states from both sides of the Fermi energy. For this reason, the elec-tronic chemical potential moves toward higher energies
0 4 8 12 16 20 0.01 0.10 1.00 10.000.01 0.10 1.00 10.000.01 0.10 1.00 10.000.01 0.10 1.00 10.00 (d) Ti 50000K (c) Ti 0K (b) Cu 50000K (a) Cu 0K Tot d s f p D en si ty o f S ta te s (s ta te s/ eV /a to m ) Energy [eV]
FIG. 4. (color online). Evolution of the total DOS and spdf -components of the DOS for Cu at 0K (a) and 5×104
K (b), and for Ti at 0K (c) and 5 × 104
K (d). Dashed lines correspond to Fermi levels.
in the case of Ni, Cu and Au, while it is displaced to-ward lower energies in the case of Al, Ti and W. In the present calculations, the Tedependence of the DOS
pro-duces important shifts of the d-block. Since the d-block concentrates electronic states, µ(Te) is also strongly
af-fected by these shifts. In Fig. 5, the relative changes of the electronic chemical potential is shown for all studied materials.
One important observation has to be made here. Pre-vious works19reported evolutions of T
edependent
prop-erties computed from DOS performed at Te= 0K. In the
present case, this approach takes into consideration the relaxion of DOS with Te, and changes in the electronic
structure are impacting the electronic chemical potential. As a general note, one can observe that trends are similar between the temperature evolution of µ derived from Te
dependent DOS and µ originating from Te = 0K DOS.
The agreement is very good in the case of Al, that we can attribute to weak changes of its electronic structure with Te. However, in case of Ni, Cu and Au, the increase of
µ is lowered in Tedependent DOS situation, a difference
0 2 4 6 8 10 -8 -6 -4 -2 0 2 4 µ-f [e V] Electronic temperature [104 K] Al Cu Ti Ni Au W
FIG. 5. (color online). Electronic chemical potential evolu-tion with the electronic temperature for all studied metals.
d-block is the main electron reservoir, when it is shifted toward lower energies, the electronic chemical potential accounts for this evolution and its displacement tends to follow. In the case of Cu for example, at 5 × 104K, µ
from Te dependent DOS is decreased by 2 eV compared
to µ from Te= 0K DOS. For W and Ti, the decrease of µ
is also lowered in Tedependent DOS. This is attributed
to the shift of the d-block toward higher energies, which is accommodated by µ. This indicates the importance of considering Te effects on band structure while regarding
the evolution of electronic populations, with direct con-sequences on the determination of free electron numbers.
IV. FREE ELECTRON NUMBERS
The number of free electrons is a quantity difficult to define since the quality of being “free” is elusive. In met-als, electrons are implicitly considered as free if they be-long to orbitals having the highest main quantum num-ber in the atomic electronic configuration (EC).39 For
example, EC of Al and Cu are respectively 3s23p1 and
3d104s1, and their corresponding number of free electrons
are 3 and 1, respectively. The highest main quantum number is chosen as it characterizes the diffuse and over-lapping nature of the orbitals. Hence, electrons belong-ing to those orbitals are assumed to be free of movbelong-ing in a large space and by extension in the whole metal, with trajectories limited by collisions and with parabolic dispersion laws. However, the use of atomic electronic configurations induces a degree of incertitude when ap-plied to condensed phases. In addition, Ne determined
from EC does not allow any change while Te increases.
As a consequence, and motivated by the necessity of giv-ing a certain evaluation of the free electron quantity, we computed the number of free electrons directly from the electronic structures of metals.
In order to improve the determination of Ne, we have
to distinguish electrons belonging to localized states (as-sumed to be non-free) from those belonging to delocalized states (considered as free). For this, it is important to
determine which are the localized states in the density of states. d-orbitals from electronic configuration of tran-sition metals defined by (n-1)dxnsy are less diffuse than
sp-orbitals due to lower main quatum number. Conse-quently, they overlap less and thus interact less than sp-orbitals. The resulting d-bands produce a characteristic d-block of very high density of electronic states. At the opposite, sp-bands are delocalized and generate sp-bands of low density with a square root distribution, similar to Al (see Fig. 1). The difference of density between lo-calized d-block and delolo-calized sp-bands is large and it is then easy to distinguish them in the DOS. Using a method to remove localized states from the DOS, one can compute free electron numbers according to the pre-vious description, i.e. electrons occupying delocalized electronic states only. Accordingly, the number of free electrons is given by the integration of the DOS weighted by the Fermi-Dirac distribution:
Ne=
Z ∞ 0
gdeloc(ε, Te)f (ε, µ, Te) dε, (2)
with gdeloc being the delocalized part of the density of
states only.
To remove localized states from the DOS, several meth-ods can be used. One of the most simple implies to fit the density of states to a curve having a square root shape, as this will artificially remove the high density of states of the d-block.40 The square root shape is chosen since
it is the distribution of the density of electronic states adopted by a free electron like metal, as observed for Al on Fig. 1a. Considering the electronic structure of d-band metals, square root shape is thus a criterium for indentifying delocalized states. Here, in order to keep all DOS subtleties, this square root fit is only used to replace the d-block, resulting in a DOS of delocalized states, that can be written as:
gdeloc(ε) = g(ε) − [g(ε) − α√ε]dblock, (3)
where α√ε is the fit of the DOS. The correction is only applied to the energy range containing the d-block. Fig. 6 exemplifies the whole process in the case of the DOS of Cu, indicating the fit results at 5 × 104K. By proceeding
0 5 10 15 20 25 30 35 0.01 0.1 1 10 Cu, FCC, 50000K D O S [s ta te s/e V/a to m ] Energy [eV]
FIG. 6. (color online). The blue curve is the density of delo-calized electronic states for Cu at 5 × 104
K. The empty red dots show the square root fit of the DOS and the dotted black curve represents the removed part of the initial DOS, consti-tuted of the non-contributing localized d-block. The hatched part refers to the free electrons and is computed from the in-tegration over the Fermi-Dirac distribution corresponding to Eq. (2).
be surprizing that, despite a strong localization charac-ter, part of d-electrons have an ability to be mobile.
Once they are only made of delocalized states, the DOS can now be integrated and Necan be deduced. The
num-ber of free electrons per atom for all considered metals is presented in Fig. 7. It shows strong variations as Te increases, exept in the case of Al, where this
num-ber remains constant with Ne = 3.0 free electrons per
atom. This is an expected free electron behavior since excited electrons are leaving delocalized states to reach other delocalized states. For Ni, Cu and Au, the localized d-block can be considered as a reservoir of non-free elec-trons susceptible to be depopulated with Te, depending
on the relative location of the Fermi energy. Then, non-free electrons from the localized d-block reach delocalized states and become free, which leads to an increase of Ne
with Te. In the case of Ti and W, the partially occupied
d-block plays an ambivalent role. At low Te the bottom
part of the d-block is filled of non-free electrons while the top part is empty but consists of localized states that can potentially trap excited electrons. As a consequence, Ne
remains constant or slightly decreases at low electronic temperatures. However, at temperatures above 104K, N
e
increases as in the case of Ni, Cu and Au, by populating delocalized states of higher energy.
The typical values of Nededuced only from electronic
configurations of isolated atoms are independent on the electronic temperature. They are provided in Table II alongside with the Neobtained by the present approach
at 0K. As already mentioned, we obtained a good agree-ment in case of Al since the electronic structure is only made of delocalized states and thus Neremains constant
and equal to the number of electrons from EC. However, differences can be large with respect to other metals, es-pecially since electronic structure of condensed phases allows the transfer of electrons between bands. They also come from the fact that we considered the d-band as partially delocalized. We observed that the free
elec-0 2 4 6 8 10 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 N e [p er a to m ] Electronic temperature [104 K] Al Cu Ti Ni Au W
FIG. 7. (color online). Evolution of the number of free elec-trons per atom (Ne) with the increase of Te, for the considered
metals.
TABLE II. Number of free electrons, from Ref. [39] and from this work obtained at Te = 0K.
Ne Al Ni Cu Au Ti W
Ref. [39] 3 2 1 1 2 2 This work 3.0 1.4 1.9 2.4 1.4 2.2
tron numbers calculated here approach values from other estimations.10,41 In case of Au, the present N
e is higher
than the value typically used. This is attributed to differ-ent Fermi energies, almost twice higher in our case (10.2 eV) than the value generally considered in the litterature (5.5 eV).39 It should also be noted that prior hypothesis
concerning the charge density led to compensating cor-rections on other electron parameters related to classical models (Drude), via e.g. the effective mass used for Au.42
V. ENERGETIC AND TRANSPORT PARAMETERS
In order to investigate to what level the band struc-ture dependence on the excitation degree (via Te) affects
macroscopic transient characteristics and physical quan-tities, we evaluated thermodynamic properties of elec-trons when Te differs from Ti. Such excitation
charac-ter corresponds to ultrashort pulse laser irradiation of metal free surfaces, where the spatio-temporal evolution of the electron distribution and its return to equilibrium is calculated by a Boltzmann formalism18or by the two
temperature model (electronic thermalization assumed). We recall that the TTM describes the energy evolution of electrons and ions subsytems using two diffusion equa-tions coupled by an electron-phonon transfer term.13The
electronic thermal energy gain is connected to the tem-perature by the electron specific heat Ce. An accurate
kinetics of the material when it is included in classical molecular dynamics simulations or in two temperatures hydrodynamics approaches.6,43,44 The electronic
contri-bution to Pe results directly from the electron heating,
depending thus on Te but also on the density of states.
Consequently, swift matter dynamics start especially due to the pressure gradient generated in the electron system. The key parameters to correctly reproduce the ultrafast nonequilibrium evolution of the material are then based on an electronic equation of states, connecting electronic specific heat Ce and pressure Pe with free electron
den-sity neand temperature. Ceand Peare generally derived
from the thermodynamic properties of an ideal Fermi gas.45–48 This work allows to insert the subtle effects of
DOS modifications allowing more accurate perspectives. Transport properties, i.e. electronic thermal and electri-cal conductivity and electron-phonon coupling strength are also important to complete the kinetic equations but they are beyond the scope of this paper and will be only briefly mentioned. We will focus here on the influence of band structure evaluation on electronic thermodynamic properties.
The electron specific heat of metals can be derived with respect to the electronic temperature by Ce = ∂E/∂Te,
where E is the internal energy of the electron system. The evolution of the specific heat under electronic exci-tation is shown in Fig. 8 where several typical behaviors are observed. For Al, Cerapidly saturates to the lowest
value of all other considered metals. The rest of met-als remains far from saturation and reaches much higher values than Al. For W and Ti, Ce exhibits a first
lev-eling from 104K to 4 × 104K and restarts to increase at
higher temperatures. At low and intermediate Tea
rela-tively good agreement is found with those obtained from g(ε) evaluated at 0K, where [∂g(ε)/∂Te]V is neglected.19
For transition metals, increasing discrepancies appear at higher temperatures, mainly due to shifts of the d-block that are highly affected by Teincrease. As expected, the
temperature dependence of Ceis linear at low electronic
temperature and tends to saturate for high Tetoward the
non-degenerate limit 3/2nelkb, where nel includes both
free electrons and part of d-electrons. As already men-tioned by Lin et al.,19 thermal excitation from the
d-band results in a positive deviation of Cefrom the linear
temperature dependence. Similar results are obtained with our temperature dependent calculations, with an additional deviation resulting from excitation of semicore electrons in case of Ti and W and high Te.
The electronic pressure Peis determined by the
deriva-tive of the electronic free energy F with respect to volume as Pe= −∂F/∂V = −∂E/∂V +Te∂S/∂V , where S is the
entropy of the system. The last term corresponding to the entropy contribution to the pressure has been shown to be largely dominant in this range of Te.30Peevolution
for the different metals is plotted in Fig. 9 and shows that Peincreases rapidly as Te2 for low excitation then scales
as Te for higher temperatures. At 2.5 × 104K, the
elec-tronic pressure is in the order of tens of GPa, and exceeds
0 2 4 6 8 10 0 2 4 6 8 10 C e [1 0 6 J m -3 K -1 ] Electronic temperature [104 K] Al Cu Ti Ni Au W
FIG. 8. (color online). Evolution of the electronic heat capac-ity with respect to the electronic temperature for the studied metals.
100 GPa at 5 × 104K for all metals except Al. Finally, at
105K more than 300 GPa are reached in case of Ni, Cu,
Au and W while it approaches a level of 200 GPa in case of Al and Ti. The fast increase of the electronic pressure is of interest since it is likely impacting the stability or the properties of materials and their evolution upon ex-citation, notably the initial steps of the thermodynamic trajectories. This strong increase of Pe with Te comes
from the occupation of the high energy states by the elec-trons, which is governed by the electronic structure and by entropic changes with the electronic temperature.
In order to exhibit band structure effects and to test our free electron approach, we renormalized Pe with
re-spect to the free electron gas pressure limit nekbTe. The
ratio is plotted in the inset of Fig. 9. At low electronic temperatures, degeneracy and band structure effects are dominating and the curves are far from the value of unity, that would characterize an ideal non-degenerated elec-tron gas behavior. However, at higher elecelec-tronic temper-atures, curves tend to saturate at the value of 1.0, which indicates that the free electron numbers we have derived are consistently characterizing the electronic pressure of the system. This asymptotic behavior is not achieved using constant values of ne given by Ref. [39]. On an
other hand, the effect of the entropy contribution on the renormalized Pe is enhanced. The entropy contribution
reflects the electronic disorder centered around the Fermi level. It comes from a compromise between the number of available electronic states and the number of electrons allowed to fill these states. In this context, when the Fermi level is within the d-block, the entropy effect is stronger, as for Ti and W, than when it is located some-where else in the DOS (as in the case of Ni, Cu, Au). Al shows the lowest renormalized values due to lowest density of states.
0 2 4 6 8 10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0 2 4 6 8 10 0.1 1 Pe [1 0 2 G Pa ] Al Cu Ti Ni Au W Te [104 K] Pe /( ne kb Te ) Electronic temperature [104 K]
FIG. 9. (color online). Evolution of the electronic pressure with the electronic temperature for all the studied elements. Inset: renormalization of Pe with respect to the free electron
gas pressure limit nekbTe.
transitions, the transport model should take into account the observed changes in band structure. In a first ap-proach, transport properties can be estimated based on classical formulas but including the evolution of ne with
Te. In this way, keconductivity can be estimated roughly
based on the relation derived from the kinetic equation between the electron heat capacity and ke.49 If τe is
the relaxation time between collisions and ve the
aver-age electron velocity, then the mean free path le= veτe
allows to make the standard connection which writes ke(Te) = 13Ce(Te)leve. Concerning the laser absorption,
insights from the classical formalism of an electron in an optical electric field, i.e. the Drude-Lorentz model, can also be obtained from the calculated properties. Whereas the energy levels and their own occupation define the re-sponse of the material to applied optical fields, it is pos-sible to describe the dielectric function by intraband and interband contributions. The optical properties are re-lated to the structure and the electronic configuration of the material and the interband is highly sensitive to the DOS derived in section III. On the other hand, the intra-band part (Drude) only depends on the number of free carriers per atom and on an effective momentum scat-tering time τ as σ = nee2τ /m(1 − iωτ), where ω is the
laser angular frequency. In this way, it becomes possi-ble to make reasonapossi-ble deductions about excitation when optical property changes are measured.50 These insights
could become more accurate if we consider informations based on electronic structure calculations, with higher computational costs in this case.51Simpler models based
on neand interband transition from filled to empty states
could be useful as a complementary approach, with nev-ertheless a lower accuracy.
VI. CONCLUSION
In the present study, Tedependent density functional
calculations were performed on a representative range of metals: Al, Ni, Cu, Au, Ti and W, with simple and transition character. Electronic temperatures from 10−2 to 105K were used to evaluate electronic properties in
nonequilibrium conditions.
In a first step DOS modifications with Teare discussed.
It is shown that almost all bands are affected by energy shifts, but the most affected states are those involving the localized part of the d-band which characterizes the tran-sition metals. Shifts towards lower energies and shrink-ing are observed for filled or almost filled d-block metals, illustrated by Ni, Cu and Au. Shifts towards higher en-ergies and extensions for partially filled d-block, with Ti and W as examples. All these modifications are explained by evolutions of electron-ion effective potential that re-sult from variation of the electronic screening generated by changes of the electronic occupation of the localized d-block. This was validated by two consistent approaches, the evolution of the number of d-electrons and the mod-ification of the Hartree energies with Te.
Changes of the DOS with electronic temperatures im-pact electronic properties, like the electronic chemical potential and the electronic heat capacities that are dis-cussed and compared to previous calculations performed at Te = 0K. A good agreement is obtained at low and
intermediate temperatures, while an increasing discrep-ancy is observed when shifts within the electronic struc-tures become stronger at high temperastruc-tures. The con-cept of electronic pressures is also addressed, with pres-sures rapidly reaching high values, of tens or hundreds of GPa, questioning material stability as Te increases.
Free electron numbers, dependent on the electronic temperature, are also computed from DOS. They are de-fined as belonging to delocalized states characterized by density of states having a square root energy dependence. As expected, for a free electron like metal as Al, this num-ber remains constant. However, Ne always increases for
transition metals, with specific behaviors depending of the filling degree of the d-block. Ni, Cu and Au exhibit a Nethat rapidly increases with Tewhile a small lag is
ob-served for Ti and W, with an increase at higher tempera-tures. At high temperatures, these free electron numbers are found consistent with Pe in an ideal gas limit.
Finally transport properties were addressed via the evolution of the electronic heat capacity and the elec-tronic pressure with Te. The electronic heat capacity
VII. AKNOWLEDGMENTS
We acknowledge Marc Torrent for providing efficient PAW atomic data. This work was supported by the ANR project DYLIPSS (ANR-12-IS04-0002-01) and by the LABEX MANUTECH-SISE (ANR-10-LABX-0075)
of the Universit´e de Lyon, within the program ”In-vestissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Part of the numerical calculations has been performed using resources from GENCI, project gen7041.
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